Proof of the Simon-Ando Theorem
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- by D. J. Hartfiel PDF
- Proc. Amer. Math. Soc. 124 (1996), 67-74 Request permission
Abstract:
In 1961, Simon and Ando wrote a classical paper describing the convergence properties of nearly completely decomposable matrices. Basically, their work concerned a partitioned stochastic matrix e.g. \[ A = \begin {bmatrix} A_1&E_1\ E_2&A_2\end {bmatrix}\] where $A_1$ and $A_2$ are square blocks whose entries are all larger than those of $E_1$ and $E_2$ respectively.
Setting \[ A^k=\begin {bmatrix} A^{(k)}_1&E^{(k)}_1\ E^{(k)}_2&A^{(k)}_2\end {bmatrix},\] partitioned as in $A$, they observed that for some, rather short, initial sequence of iterates the main diagonal blocks tended to matrices all of whose rows are identical, e.g. $A^{(k)}_1$ to $F_1$ and $A^{(k)}_2$ to $F_2$. After this initial sequence, subsequent iterations showed that all blocks lying in the same column as those matrices tended to a scalar multiple of them, e.g. \[ \lim _{k\to \infty }A^k=\begin {bmatrix} \alpha F_1&\beta F_2\ \alpha F_1&\beta F_2\end {bmatrix}\] where $\alpha \geq 0$ and $\beta \geq 0$.
The purpose of this paper is to give a qualitative proof of the Simon-Ando theorem.
References
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- Herbert A. Simon and Albert Ando, Aggregation of variables in dynamic systems, Econometrica 29 (1961), 111–138.
Additional Information
- D. J. Hartfiel
- Affiliation: Department of Mathematics, Texas A & M University, College Station, Texas 77843
- Email: hartfiel@math.tamu.edu
- Received by editor(s): February 9, 1994
- Received by editor(s) in revised form: August 18, 1994
- Communicated by: Joseph S. B. Mitchell
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 67-74
- MSC (1991): Primary 15A51, 15A48
- DOI: https://doi.org/10.1090/S0002-9939-96-03033-X
- MathSciNet review: 1291772